I was working for a temporary agency for 4-5 months, after being asked to stay permanently on, the company did a drug screen and failed for THC Marijuana, and no longer employed with actual company (not temp agency). Do I need to include this on employment history and resume. Will employer find out this information.
A few things here.
You say that you failed a drug test from the client company, not the temp agency you work through, and only after they offered a permanent position. Are you still employed with the temp agency?
As DarkCygnus comments, its preferred to include experiences of at least 6 months, but a 4-5 month gap would be noticed. And as Mister Positive notes, it could be discovered in a background check. Dukeling's comment makes a good point as well.
So, what do you do? Well, it doesn't sound like you were actually terminated from the job, but simply not extended a permanent position for not meeting the company's drug policy. If you are still employed through the temp agency this wouldn't be much issue. If you aren't employed through the agency, this still isn't much of a problem either.
Your resume only needs to include dates and brief description of job. In this case, you could simply state you worked at Acme Temp Agency from Jan 1 - May 31. List some details of the jobs you did. You can include that you worked for Acme but performed work at Widget Co., but if prospective employer has contacts there they could inquire about you. An interviewer may question about the work you did, and you can just tell them that for privacy you wish not to name (or aren't allowed to) the clients you worked at. If you do tell them, when asked in an interview why you weren't hired on permanently, you simply state you weren't extended a permanent position.